Celebratio Mathematica

John Mor­gan and I both at­ten­ded Rice Uni­versity in dis­joint but ad­ja­cent four year peri­ods. It was Rice In­sti­tute of Tech­no­logy when I entered in 1959. It be­came Rice Uni­versity halfway through my time there. Rice was ex­tremely de­mand­ing for me but ap­par­ently less so for John Mor­gan. John was tak­ing grad courses most of the way through, gain­ing his PhD in his fifth year at Rice after en­ter­ing there from high school in East Texas. I had star­ted Rice In­sti­tute as a chem­ic­al en­gin­eer and I liked phys­ic­al chem­istry very much, es­pe­cially ther­mo­dy­nam­ics. However I changed to math­em­at­ics after learn­ing about the Riemann map­ping the­or­em in the com­plex vari­able part of the second year math course and about the ran­dom path char­ac­ter­iz­a­tion of har­mon­ic func­tions.

John and I first met at Prin­ceton when both had PhDs. My work at that point de­pended on some di­li­gence and much good for­tune. John was in­ter­ested and de­voured the ideas all the way down to the ba­sics. The work be­came known, es­pe­cially when the odd prime ob­struc­tions de­pended on be­ing equi­val­ent to the odd prime part of K the­ory. Real and com­plex K-the­ory were each hot top­ics at the time be­cause of the Atiyah–Sing­er in­dex the­or­em. Al­though my work gained at­ten­tion, I no­ticed that, to my know­ledge, only two oth­er people seemed to un­der­stand it com­pletely, and one of those two was John Mor­gan.

John offered to write care­ful notes of my lec­tures at Prin­ceton. This of­fer I ac­cep­ted gladly since I had trouble writ­ing any­thing bey­ond out­lines of the ideas after (i) lec­tur­ing on them to ex­perts, and (ii) after I felt I had un­der­stood things quite well as a geo­met­ric to­po­lo­gist. When my ad­visor Bill Browder learned of this ar­range­ment, he scol­ded me for dis­tract­ing John from his own re­search. In ret­ro­spect I might have countered that John’s mod­us op­erandi is not only to un­der­stand the­or­ems and their proofs but also to op­tim­ize the con­cepts, find bet­ter lem­mas, in short to make everything as simple as pos­sible and, most of all, as clear as pos­sible. However I fol­lowed my thes­is ad­visor’s ad­vice, and the pro­ject with John was aban­doned. The res­ult was that this the­ory (about clas­si­fy­ing PL man­i­folds in a ho­mo­topy type by fi­nitely many nu­mer­ic­al in­vari­ants) be­came ob­scure (en­tirely my fault).

A few years later John and I teamed up to de­vel­op the the­ory of nu­mer­ic­al in­vari­ants fur­ther at the prime two [1]. Us­ing this meth­od, we showed that to­po­lo­gic­al and PL man­i­folds had at the prime two ca­non­ic­al 2-in­teg­ral Hirzebruch \( L \)-classes. (Note, however, that these classes for smooth man­i­folds are de­scribed by Pontry­agin classes, which are in­teg­ral classes, times frac­tions with odd de­nom­in­at­ors like \( 1/3, 1/45, \dots\, \), and that the con­crete mean­ing of this arith­met­ic fact ex­tends to PL and to­po­lo­gic­al man­i­folds was one of two main the­or­ems in the An­nals pa­per [1] with John. The second point was about sur­gery the­ory and the mys­ter­i­ous di­vi­sion of a sig­na­ture by 8.) There were tech­nic­al as­pects of this finely writ­ten pa­per by John, like the com­mut­at­ive square in­volving coef­fi­cients in the ra­tion­als and the ra­tion­als mod in­tegers which greatly cla­ri­fied the dis­cus­sion. This square is also use­ful at the present time. For ex­ample, re­cent Stony Brook PhD Ji­ahao Hu ex­ten­ded in his thes­is the idea and real­ity of a com­plete fi­nite set of nu­mer­ic­al in­vari­ants for ele­ments in real K-the­ory (in­stead of just for com­plex K-the­ory and only just for odd primes for real K-the­ory). This la­cuna in our un­der­stand­ing had been open for over 50 years.

After this John worked on mul­tiple areas between to­po­logy and geo­metry some­times with the same mod­us op­erandi, in some cases sim­pli­fy­ing and op­tim­iz­ing state­ment and proof, in ef­fect ex­pand­ing out the cut­ting edge of the field. He did this in act­ive areas like gauge the­ory, sym­plect­ic to­po­logy and geo­met­riz­a­tion in three di­men­sions, and in oth­er areas, as well, by plow­ing anew a field that had lain dormant for dec­ades. The ex­ample I am cur­rently study­ing is his re­vis­it­ing activ­ity in the older work of Serre and Tits about groups act­ing on graphs. This “reawaken­ing” work of John Mor­gan with Peter Shalen has been im­port­ant for un­der­stand­ing the geo­metry of groups in the sense of Gro­mov.

There is a third type of ex­ample of John’s re­search: In the ra­tion­al ho­mo­topy the­ory defined by dif­fer­en­tial forms the pos­sib­il­ity arose to ap­ply this the­ory to Kähler man­i­folds be­cause look­ing at the closed holo­morph­ic forms showed such a man­i­fold must sat­is­fy in­fin­itely many high­er-or­der van­ish­ing con­di­tions. After shar­ing this thought with Phil Grif­fiths, he, John and I star­ted to work on this in the early 1970s. The full van­ish­ing the­or­em (now called form­al­ity) de­fied proof for some time. Even­tu­ally Pierre De­ligne, who had joined the pro­ject, noted that for al­geb­ra­ic Kähler man­i­folds a weight grad­ing ar­gu­ment would im­ply the res­ult if one of the main Weil con­jec­tures were true. (That con­jec­ture was about the ei­gen val­ues of Frobeni­us. Pierre was hard at work try­ing to prove this and suc­ceeded not long after.) In the mean­time John and Phil wrote up notes about dif­fer­en­tial form ho­mo­topy the­ory, and, work­ing to­geth­er di­li­gently, ob­tained a proof. (Later I found a dif­fer­ent short proof of the form­al­ity, and in­cluded it in a de­cent ac­count of the en­tire dif­fer­en­tial form ho­mo­topy the­ory [2].) Then here comes the third type of ex­ample of John’s re­search. John con­tin­ued us­ing De­ligne’s al­geb­ra­ic ideas to study with dif­fer­en­tial form ho­mo­topy the­ory the much harder prob­lem of the nature of the ra­tion­al ho­mo­topy type of open af­fine al­geb­ra­ic vari­et­ies. He man­aged to prove a re­mark­able prop­erty: The in­teg­ral ho­mo­topy type, if simply con­nec­ted, has enough con­tinu­ous self map­pings which are ra­tion­al equi­val­ences to make any in­teg­ral co­homo­logy class ar­bit­rar­ily di­vis­ible in its in­teg­ral co­homo­logy group. This prop­erty called pos­it­ive weights shows any ob­struc­tion the­ory prob­lem over \( Z \) with fi­nite ob­struc­tions can be suc­cess­fully worked around by in­duct­ively pulling back the prob­lem by the now known self map­pings.

Fi­nally, when Jim Si­mons star­ted or­gan­iz­ing the star­tup of the Si­mons Cen­ter for Geo­metry and Phys­ics (Stony Brook Uni­versity), a num­ber of not­able people, in­clud­ing John Mor­gan, were in­vited to join a meet­ing os­tens­ibly to ask their ad­vice about such an en­deavor. As John left the meet­ing he dropped a last re­mark: “I don’t un­der­stand what string the­ory means in phys­ics but in math­em­at­ics its just the thing, won­der­ful.” A con­sensus of math and phys­ics people emerged to in­vite him to be the first dir­ect­or of the SCGP. I called him to feel it out. He said on the phone he was not in­ter­ested be­cause he was just fin­ish­ing his turn as Chair at Columbia and couldn’t wait to get back to re­search. I wasn’t sur­prised, and the call ended.

Fi­nally, when Jim Si­mons star­ted or­gan­iz­ing the star­tup of the Si­mons Cen­ter for Geo­metry and Phys­ics (Stony Brook Uni­versity), a num­ber of not­able people, in­clud­ing John Mor­gan, were in­vited to join a meet­ing os­tens­ibly to ask their ad­vice about such an en­deavor. As John left the meet­ing he dropped a last re­mark: “I don’t un­der­stand what string the­ory means in phys­ics but in math­em­at­ics it’s just the thing, won­der­ful.” A con­sensus of math and phys­ics people emerged at the meet­ing to in­vite John Mor­gan to be the first dir­ect­or of the SCGP. I called him to feel out his po­s­i­tion. He said on the phone he was not in­ter­ested be­cause he was just fin­ish­ing his turn as Chair at Columbia and couldn’t wait to get back to re­search. I wasn’t sur­prised, and the call ended.

However the next day John called back to cla­ri­fy something about this pos­sib­il­ity: “Does the job de­scrip­tion re­quire fun­drais­ing?” I answered, “I thought not,” and the call ended.

John be­came the found­ing Dir­ect­or of the Si­mons Cen­ter for Geo­metry and Phys­ics, for the next sev­en plus years. He did a won­der­ful job.